 reserve x,y,X,Y for set;
reserve G for non empty multMagma,
  D for set,
  a,b,c,r,l for Element of G;
reserve M for non empty multLoopStr;
reserve H for non empty SubStr of G,
  N for non empty MonoidalSubStr of G;

theorem
  for H1,H2 being non empty MonoidalSubStr of M st
    the carrier of H1 = the carrier of H2 holds
      the multLoopStr of H1 = the multLoopStr of H2
proof
  let H1,H2 be non empty MonoidalSubStr of M such that
A1: the carrier of H1 = the carrier of H2;
  reconsider N1 = H1, N2 = H2 as SubStr of M by Th21;
A2: un(H1) = un(M) & un(H2) = un(M) by Def25;
  the multMagma of N1 = the multMagma of N2 by A1,Th26;
  hence thesis by A2;
end;
